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In the philosophy of mathematics, the abstraction of actual infinity, also called completed infinity, [1] involves infinite entities as given, actual and completed objects. The concept of actual infinity has been introduced in mathematics near the end of the 19th century by Georg Cantor , with his theory of infinite sets , later formalized into ...
In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object. The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets.
Aristotle proposed a three-part structure for souls of plants, animals, and humans, making humans unique in having all three types of soul. Aristotle's psychology, given in his treatise On the Soul (peri psychēs), posits three kinds of soul ("psyches"): the vegetative soul, the sensitive soul, and the rational soul. Humans have all three.
But since Aristotle holds that such treatments of infinity are impossible and ridiculous, the world cannot have existed for infinite time. [9] Philoponus's works were adopted by many; his first argument against an infinite past being the "argument from the impossibility of the existence of an actual infinite", which states: [10]
Aristotle especially promoted the potential infinity as a middle option between strict finitism and actual infinity (the latter being an actualization of something never-ending in nature, in contrast with the Cantorist actual infinity consisting of the transfinite cardinal and ordinal numbers, which have nothing to do with the things in nature):
Temporal finitism is the doctrine that time is finite in the past. [clarification needed] The philosophy of Aristotle, expressed in such works as his Physics, held that although space was finite, with only void existing beyond the outermost sphere of the heavens, time was infinite.
Aristotle here says the only type of infinity that exists is the potentially infinite. Aristotle characterizes this as that which serves as "the matter for the completion of a magnitude and is potentially (but not actually) the completed whole" (207a22-23). The infinite, lacking any form, is thereby unknowable.
In philosophy and theology, infinity is explored in articles under headings such as the Absolute, God, and Zeno's paradoxes. In Greek philosophy, for example in Anaximander, 'the Boundless' is the origin of all that is.